This study develops a quantum switching device with fully nonblocking properties. Although previous studies have also presented quantum-based solutions for the blocking problem, the proposed schemes are characterized by an increased packet loss, a large number of quantum SWAP gates and an increased propagation delay time complexity. The current study overcomes these drawbacks by designing an N \times N fully nonblocking quantum switch, in which the packet payload is passed through quantum SWAP gates while the packet header is passed through quantum control gates designed by applying a modified quantum Karnaugh mapping method. The allocation of quantum SWAP gates to the different layers within the switch is solved using a Perfect Matching in Complete Graph (PMiCG) algorithm with a time complexity of {\rm O}(N!/(2^{N/2}(N/2)!)). A symmetry-based heuristic method is proposed to reduce the time complexity of the search process for all the perfect matching pairs to a time complexity of {\rm O}(N^{2}). The performance of the proposed quantum switch is compared with that of a quantum self-routing packet switch and a quantum switching/quantum merge sorting scheme, respectively, in terms of the hardware complexity, the propagation delay time complexity, the auxiliary qubit complexity, and the packet loss probability.